Mixing in confined stratified aquifers

Spatial variability in a flow field leads to spreading of a tracer plume. The effect of microdispersion is to smooth concentration gradients that exist in the system. The combined effect of these two phenomena leads to an 'effective' enhanced mixing that can be asymptotically quantified by an effective dispersion coefficient (i.e. Taylor dispersion). Mixing plays a fundamental role in driving chemical reactions. However, at pre-asymptotic times it is considerably more difficult to accurately quantify these effects by an effective dispersion coefficient as spreading and mixing are not the same (but intricately related). In this work we use a volume averaging approach to calculate the concentration distribution of an inert solute release at pre-asymptotic times in a stratified formation. Mixing here is characterized by the scalar dissipation rate, which measures the destruction of concentration variance. As such it is an indicator for the degree of mixing of a system. We study pre-asymptotic solute mixing in terms of explicit analytical expressions for the scalar dissipation rate and numerical random walk simulations. In particular, we divide the concentration field into a mean and deviation component and use dominant balance arguments to write approximate governing equations for each, which we then solve analytically. This allows us to explicitly evaluate the separate contributions to mixing from the mean and the deviation behavior. We find an approximate, but accurate expression (when compared to numerical simulations) to evaluate mixing. Our results shed some new light on the mechanisms that lead to large scale mixing and allow for a distinction between solute spreading, represented by the mean concentration, and mixing, which comes from both the mean and deviation concentrations, at pre-asymptotic times.